View Item 

Show simple item record

Translation surfaces and periods of meromorphic differentials
dc.contributor.authorChenakkod, Shabarish
dc.contributor.authorFaraco, Gianluca
dc.contributor.authorGupta, Subhojoy
dc.date.accessioned2022-04-08T18:03:22Z
dc.date.available2023-05-08 14:03:20en
dc.date.available2022-04-08T18:03:22Z
dc.date.issued2022-04
dc.identifier.citationChenakkod, Shabarish; Faraco, Gianluca; Gupta, Subhojoy (2022). "Translation surfaces and periods of meromorphic differentials." Proceedings of the London Mathematical Society 124(4): 478-557.
dc.identifier.issn0024-6115
dc.identifier.issn1460-244X
dc.identifier.urihttps://hdl.handle.net/2027.42/171999
dc.description.abstractLet S$S$ be an oriented surface of genus g$g$ and n$n$ punctures. The periods of any meromorphic differential on S$S$, with respect to a choice of complex structure, determine a representation χ:Γg,n→C$chi :Gamma _{g,n} rightarrow mathbb {C}$ where Γg,n$Gamma _{g,n}$ is the first homology group of S$S$. We characterise the representations that thus arise, that is, lie in the image of the period map Per:ΩMg,n→Hom(Γg,n,C)$textsf {Per}:Omega mathcal {M}_{g,n}rightarrow textsf {Hom}(Gamma _{g,n}, {mathbb {C}})$. This generalises a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on S$S$ with the prescribed holonomy χ$chi$. Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.
dc.publisherAmer. Math. Soc
dc.publisherWiley Periodicals, Inc.
dc.titleTranslation surfaces and periods of meromorphic differentials
dc.typeArticle
dc.rights.robotsIndexNoFollow
dc.subject.hlbsecondlevelMathematics
dc.subject.hlbtoplevelScience
dc.description.peerreviewedPeer Reviewed
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/171999/1/plms12432.pdf
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/171999/2/plms12432_am.pdf
dc.identifier.doi10.1112/plms.12432
dc.identifier.sourceProceedings of the London Mathematical Society
dc.identifier.citedreferenceM. Kontsevich and A. Zorich, Connected components of the moduli spaces of abelian differentials with prescribed singularities, Invent. Math. 153 ( 2003 ), no. 3, 632 – 678.
dc.identifier.citedreferenceS. Baba and S. Gupta, Holonomy map fibers of CP 1 ${mathbb {CP}}^1$ ‐structures in moduli space, J. Topol. 8 ( 2015 ), no. 3, 691 – 710.
dc.identifier.citedreferenceM. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. Möller, Strata of k $k$ ‐differentials, Algebr. Geom. 6 ( 2019 ), 196 – 233.
dc.identifier.citedreferenceM. Bainbridge, C. Johnson, C. Judge, and I. Park, Haupt’s theorem for strata of abelian differentials, arxiv:2002.12901, 2020.
dc.identifier.citedreferenceD. Gallo, M. Kapovich, and A. Marden, The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. (2) 151 ( 2000 ), no. 2, 625 – 704. MR 1765706.
dc.identifier.citedreferenceQ. Gendron and G. Tahar, Isoresidual fibration and resonance arrangements, arxiv:2007.14872, 2020.
dc.identifier.citedreferenceQ. Gendron and G. Tahar, Différentielles abéliennes á singularités prescrites, J. Éc. polytech. Math. 8 ( 2021 ), 1397 – 1428.
dc.identifier.citedreferenceS. Ghazouani, Mapping class group dynamics and the holonomy of branched affine structures, Math. Z. 289 ( 2018 ), no. 1–2, 1 – 23.
dc.identifier.citedreferenceS. Gupta, Monodromy groups of C P 1 $mathbb {C}{P}^1$ ‐structures on punctured surfaces, J. Topol. 14 ( 2021 ), no. 2, 538 – 559.
dc.identifier.citedreferenceU. Hamenstädt, Ergodicity of the absolute period foliation, Israel J. Math. 225 ( 2018 ), 661 – 680.
dc.identifier.citedreferenceO. Haupt, Ein satz über die abelschen integrale, I. Gattung. Math. Zeit. 6 ( 1920 ), no. 3–4, 219 – 237.
dc.identifier.citedreferenceM. Kapovich, Periods of abelian differentials and dynamics, Dynamics: topology and numbers, Contemporary Mathematics, vol. 744, Amer. Math. Soc., Providence, RI, 2020, pp. 297 – 315.
dc.identifier.citedreferenceT. Le Fils, Periods of abelian differentials with prescribed singularities, International Mathematics Research Notices ( 2020 ), Published online at https://doi.org/10.1093/imrn/rnaa263.
dc.identifier.citedreferenceC. T. McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math. 129 ( 2007 ), 183 – 215.
dc.identifier.citedreferenceC. T. McMullen, Moduli spaces of isoperiodic forms, Duke Math. J. 163 ( 2014 ), no. 12, 2271 – 2323.
dc.identifier.citedreferenceM. Moskowitz and R. Sacksteder, An extension of a theorem of Hlawka, Mathematika 56 ( 2010 ), no. 2, 203 – 216.
dc.identifier.citedreferenceE. Pervova and C. Petronio, On the existence of branched coverings between surfaces with prescribed branch data. I, Algebr. Geom. Topol. 6 ( 2006 ), 1957 – 1985.
dc.identifier.citedreferenceS. P. Tan, Branched C P 1 $mathbb {C}mathbb {P}^1$ ‐structures on surfaces with prescribed real holonomy, Math. Ann. 300 ( 1994 ), no. 4, 649 – 667.
dc.identifier.citedreferenceF. Ygouf, A criterion for density of the isoperiodic leaves in rank 1 affine invariant suborbifolds, arXiv:2002.01186, 2020.
dc.identifier.citedreferenceK. Barański, On realizability of branched coverings of the sphere, Topology Appl. 116 ( 2001 ), 279 – 291.
dc.identifier.citedreferenceC. Boissy, Connected components of the strata of the moduli space of meromorphic differentials, Comment. Math. Helv. 90 ( 2015 ), no. 2, 255 – 286.
dc.identifier.citedreferenceG. Calsamiglia, B. Deroin, and S. Francaviglia, A transfer principle: From periods to isoperiodic foliations, arxiv:1511.07635.
dc.identifier.citedreferenceG. Faraco and S. Gupta, Monodromy groups of Schwarzian equations with regular singularities, arxiv:2109.04044, 2021.
dc.identifier.citedreferenceS. Baba, 2 π $2pi$ ‐grafting and complex projective structures with generic holonomy, Geom. Funct. Anal. 27 ( 2017 ), no. 5, 1017 – 1069.
dc.working.doiNOen
dc.owningcollnameInterdisciplinary and Peer-Reviewed


Files in this item

Name:
plms12432.pdf
Size:
6.721MB
Format:
PDF
View/Open
Name:
plms12432_am.pdf
Size:
902.5KB
Format:
PDF
View/Open

Show simple item record

Remediation of Harmful Language

The University of Michigan Library aims to describe library materials in a way that respects the people and communities who create, use, and are represented in our collections. Report harmful or offensive language in catalog records, finding aids, or elsewhere in our collections anonymously through our metadata feedback form. More information at Remediation of Harmful Language.

Accessibility

If you are unable to use this file in its current format, please select the Contact Us link and we can modify it to make it more accessible to you.